Page:Newton's Principia (1846).djvu/290

 II of this Book) the moment KL of AK will be equal to $$\scriptstyle \frac{2APQ+2BA\times PQ}{Z}$$ or $$\scriptstyle \frac{2BPQ}{Z}$$, and the moment KLON of the area AbNK will be equal to $$\scriptstyle \frac{2BPQ\times LO}{Z}$$ or $$\scriptstyle \frac{BPQ\times BD^{3}}{2Z\times CK\times AB}$$.

1. Now if the body ascends, and the gravity be as AB² + BD², BET being a circle, the line AC, which is proportional to the gravity, will be $$\scriptstyle \frac{AB^{2}+BD^{2}}{Z}$$, and DP² or AP² + 2BAP + AB² + BD² will be AK $$\scriptstyle \times$$ Z + AC $$\scriptstyle \times$$ Z or CK $$\scriptstyle \times$$ Z; and therefore the area DTV will be to the area DPQ as DT² or DB² to CK $$\scriptstyle \times$$ Z.

2. If the body ascends, and the gravity be as AB² - BD², the line AC will be $$\scriptstyle \frac{AB^{2}-BD^{2}}{Z}$$, and DT² will be to DP² as DF² or DB² to BP² - BD² or AP² + 2BAP + AB² - BD², that is, to AK $$\scriptstyle \times$$ Z +



AC $$\scriptstyle \times$$ Z or CK $$\scriptstyle \times$$ Z. And therefore the area DTV will be to the area DPQ as DB² to CK $$\scriptstyle \times$$ Z.

3. And by the same reasoning, if the body descends, and therefore the gravity is as BD² - AB², and the line AC becomes equal to $$\scriptstyle \frac{BD^{2}-AB^{2}}{Z}$$; the area DTV will be to the area DPQ, as DB² to CK $$\scriptstyle \times$$ Z: as above.

Since, therefore, these areas are always in this ratio, if for the area